Dynamic drop breakup model

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

A fragmented liquid core is simulated by injecting large drops which break up into smaller and smaller product droplets, until the latter reach a stable condition. The primary breakup, that is, the first drop breakup after injection, is modeled by delaying the initial drop breakup in accordance with experimental correlations. The drop distortion and the breakup criterion are obtained from Taylor s drop oscillator. The properties of the product droplets are derived from principles of population dynamics and are modeled after experimentally observed droplet breakup mechanisms. 

In the CAB model the breakup condition is determined by means of the drop deformation dynamics of the standard Taylor analogy breakup model [5] (cf. TAB model above). In this approach, the drop distortion is described by a forced, damped, harmonic oscillator in which the forcing term is given by the aerodynamic droplet-gas interaction, the damping is due to the liquid viscosity and the restoring force is supplied by the surface tension. More specifically, the drop distortion is described by the deformation parameter, y = Ixjr, where x denotes the largest radial distortion from the spherical equilibrium surface, and r is the drop radius. The deformation equation in terms of the normalized distortion parameter, y, as provided in Eq. 9.29 is... 

The dynamic behavior of polymer blends under low strain has been theoretically treated from the perspective of microrheology. Table 2.3 lists a summary of this approach. These models well describe the experimental data within the range of stresses and concentrations where neither drop-breakup nor coalescence takes place. The two latter models yield similar predictions as that of Palierne. The last entry in the Table 2.3 is an empirical modification of Palieme s model by replacement of the volume fraction of dispersed phase by its efiective quantity (Eq. (2.24)), which extends the applicability of the relation up to 0 < 0.449. However, at these high concentrations the drop-drop interactions absent in the Palierne model must complicate the flow and coalescence is expected. The practical solution to the latter problem is compatibilization, but the presence of the third component in blends has not been treated theoretically. 

The idea of a dynamic fragmentation model, which calculates the characteristic melt diameter as a function of instantaneous hydrodynamic conditions, was first proposed by Camp in Ref, 30. A model using this idea was later incorporated into a version of the Thermal Explosion Analysis System (TEXAS)one-dimensional FCI code by Chu and Corradini, using an empirical correlation derived from data obtained in the FITS experiments. The fragmentation model in IFCI is a version of a dynamic fragmentation model developed by Pilch based on Rayleigh-Taylor instability theory and the existing body of gas-liquid and liquid-liquid drop breakup data. 

The foregoing example is interesting because it shows population balance models can account for the occurrence of physicochemical processes in dispersed phase systems simultaneously with the dispersion process itself. Shah and Ramkrishna (1973) also show how the predicted mass transfer rates vary significantly from those obtained by neglecting the dynamics of drop breakage. The model s deficiencies (such as equal binary breakage) are deliberate simplifications because its purpose had been to demonstrate the importance of the dynamics of dispersion processes in the calculation of mass transfer rates rather than to be precise about the details of drop breakup. 

Ibrahim et al. [12] proposed the Droplet Deformation Breakup (DDB) model, which is based on the drop s dynamics in terms of the motion of the center-of-mass of the half-droplet. It is assumed that the liquid drop is deformed due to a pure extensional flow from an initial spherical shape of radius r into an oblate spheroid having an ellipsoidal cross-section with major semi-axis a and minor semi-axis b. The internal energy of the half-drop comes from the sum of its kinetic and potential energies, E, expressed as follows ... 

Unlike in NEMD models, the microstructures emerging due to competition between the breakup and coalescence processes can be studied by using DPD modeling. For example, in Figure 26.23, the four principal mechanisms, the same as those responsible for droplets breakup [ 118,119], can be observed in DPD simulation of the R-T instability. As shown in [116,119], moderately extended drops for capillary number close to a critical value, which is a function of dynamic viscosity ratio... 

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